Reconciling common source, specific source, feature based and score based likelihood ratios

We show that the incorporation of any new piece of information allows for improved decision making in the sense that the expected costs of an optimal decision decrease (or, in boundary cases where no or not enough new information is incorporated, stays the same) whenever this is done by the appropriate update of the probabilities of the hypotheses. Versions of this result have been stated before. However, previous proofs rely on auxiliary constructions with proper scoring rules. We, instead, offer a direct and completely general proof by considering elementary properties of likelihood ratios only. We apply our results to make a contribution to the debates about the use of score based/feature based and common/specific source likelihood ratios. In the literature these are often presented as different ``LR-systems''. We argue that the difference between these is simply a matter which information is processed. There is no therefore no such thing as different ``LR-systems'', there are only differences in the processed information. In particular, despite claims to the contrary, scores can very well be used in forensic practice and we illustrate this with an extensive example in DNA kinship context.